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Math. Comput.</a> <a href="/?q=in%3A335294" title="Articles in this Issue">219, No. 20, 10401-10415 (2013)</a>. </div> <div class="abstract">Summary: Recently, machine scheduling problems with deteriorating jobs have received more attention from the scheduling research community. In this paper we consider a single-machine scheduling problem with a linear deteriorating effect and unequal release times. The objective is to minimize the total tardiness. We propose a branch-and-bound algorithm incorporating with several dominance properties and two lower bounds to search for the optimal solution. In addition, we propose a marriage in honey-bees optimization algorithm (MBO) to provide a near-optimal solution. The computational experiment is also conducted to evaluate the impacts of the parameters over the performances of the proposed algorithms.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6325715">Cited in <strong>7</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A90B35" title="MSC2020">90B35</a> </td> <td class="space"> Deterministic scheduling theory in operations research </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ascheduling">scheduling</a>; <a href="/?q=ut%3Asingle-machine">single-machine</a>; <a href="/?q=ut%3Atardiness">tardiness</a>; <a href="/?q=ut%3Adeteriorating+jobs">deteriorating jobs</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1293.90028" data-ciurl="/ci/06325715" data-biburl="/bibtex/06325715.bib" data-amsurl="/amsrefs/06325715.bib" data-xmlurl="/xml/06325715.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06325715.pdf" title="Zbl 1293.90028 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.amc.2013.04.018" aria-label="DOI for “The single-machine total tardiness problem with unequal release times and a linear deterioration”" title="10.1016/j.amc.2013.04.018">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[4]</td> <td class="space">Abdul-Razaq, T. S.; Potts, C. N.; Van Wassenhove, L. N., A survey of algorithms for the single machine total weighted tardiness scheduling problem, Discrete Applied Mathematics, 26, 235-253 (1990) · <a href="/0685.90059" class="nowrap">Zbl 0685.90059</a></td> </tr><tr> <td>[5]</td> <td class="space">Alidaee, B.; Womer, N. K., Scheduling with time dependent processing times: review and extensions, Journal of the Operational Research Society, 50, 711-729 (1999) · <a href="/1054.90542" class="nowrap">Zbl 1054.90542</a></td> </tr><tr> <td>[7]</td> <td class="space">Bachman, A.; Janiak, A., Minimizing maximum lateness under linear deterioration, European Journal of Operational Research, 126, 557-566 (2000) · <a href="/0976.90039" class="nowrap">Zbl 0976.90039</a></td> </tr><tr> <td>[8]</td> <td class="space">Bachman, A.; Janiak, A.; Kovalyov, M. Y., Minimizing the total weighted completion time of deteriorating jobs, Information Processing Letters, 81, 81-84 (2002) · <a href="/1032.68019" class="nowrap">Zbl 1032.68019</a></td> </tr><tr> <td>[9]</td> <td class="space">Bachman, A.; Cheng, T. C.E.; Janiak, A.; Ng, C. T., Scheduling start time dependent jobs to minimize the total weighted completion time, Journal of the Operational Research Society, 53, 6, 688-693 (2002) · <a href="/1059.90063" class="nowrap">Zbl 1059.90063</a></td> </tr><tr> <td>[10]</td> <td class="space">Bachman, A.; Cheng, T. C.E.; Janiak, A.; Ng, C. T., Three scheduling problems with deteriorating jobs to minimize the total completion time, Information Processing Letters, 81, 6, 327-333 (2002)</td> </tr><tr> <td>[11]</td> <td class="space">Benatchba, K.; Admane, L.; Koudil, M.; Madjour, S.; Alileche, Y., Using Bees to Solve a Data-Mining Problem Expressed as a Max-Sat One. Using Bees to Solve a Data-Mining Problem Expressed as a Max-Sat One, Lecture Notes in Computer Science, vol. 3562 (2005), Springer-Verlag</td> </tr><tr> <td>[12]</td> <td class="space">Browne, S.; Yechiali, U., Scheduling deteriorating jobs on a single processor, Operations Research, 38, 495-498 (1990) · <a href="/0703.90051" class="nowrap">Zbl 0703.90051</a></td> </tr><tr> <td>[13]</td> <td class="space">Cheng, T. C.E.; Ding, Q.; Lin, B. M.T., A concise survey of scheduling with time-dependent processing times, European Journal of Operational Research, 152, 1-13 (2004) · <a href="/1030.90023" class="nowrap">Zbl 1030.90023</a></td> </tr><tr> <td>[14]</td> <td class="space">Cheng, T. C.E.; Yang, S. J.; Yang, D. L., Common due-window assignment and scheduling of linear time-dependent deteriorating jobs and a deteriorating maintenance activity, International Journal of Production Economics, 135, 1, 154-161 (2012)</td> </tr><tr> <td>[15]</td> <td class="space">Della Croce, F.; Narayan, V.; Tadei, R., The two-machine total completion time flow shop problem, European Journal of Operational Research, 90, 227-237 (1996) · <a href="/0916.90148" class="nowrap">Zbl 0916.90148</a></td> </tr><tr> <td>[16]</td> <td class="space">Fisher, M. L., A dual algorithm for the one-machine scheduling problem, Math Programming, 11, 229-251 (1971) · <a href="/0359.90039" class="nowrap">Zbl 0359.90039</a></td> </tr><tr> <td>[17]</td> <td class="space">French, S., Sequencing and Scheduling - An Introduction to the Mathematics of the Job-Shop (1982), Ellis Horwood Limited: Ellis Horwood Limited UK · <a href="/0479.90037" class="nowrap">Zbl 0479.90037</a></td> </tr><tr> <td>[18]</td> <td class="space">Gawiejnowicz, S., Time-Dependent Scheduling (2008), Springer-Verlag Inc.: Springer-Verlag Inc. New York · <a href="/1155.90004" class="nowrap">Zbl 1155.90004</a></td> </tr><tr> <td>[19]</td> <td class="space">Inderfurth, K.; Janiak, A.; Kovalyov, M. Y.; Werner, F., Batching work and rework processes with limited deterioration of reworkables, Computers and Operations Research, 33, 1595-1605 (2006) · <a href="/1087.90008" class="nowrap">Zbl 1087.90008</a></td> </tr><tr> <td>[20]</td> <td class="space">Janiak, A.; Kovalyov, M. Y., Scheduling deteriorating jobs, (Janiak, A., Scheduling in Computer and Manufacturing Systems (2006), Warszawa: Warszawa WKL), 12-25</td> </tr><tr> <td>[21]</td> <td class="space">Koudil, M.; Benatchba, K.; Tarabet, A.; Sahraoui, E. B., Using artificial bees to solve partitioning and scheduling problems in codesign, Applied Mathematics and Computation, 186, 1710-1722 (2007) · <a href="/1128.90027" class="nowrap">Zbl 1128.90027</a></td> </tr><tr> <td>[22]</td> <td class="space">Koulamas, K., The total tardiness problem: review and extensions, Operations Research, 42, 1025-1041 (1994) · <a href="/0824.90083" class="nowrap">Zbl 0824.90083</a></td> </tr><tr> <td>[23]</td> <td class="space">Koulamas, K., The single-machine total tardiness scheduling problem: review and extensions, European Journal of Operational Research, 202, 1-7 (2010) · <a href="/1175.90174" class="nowrap">Zbl 1175.90174</a></td> </tr><tr> <td>[24]</td> <td class="space">Kunnathur, A. S.; Gupta, S. K., Minimizing the makespan with late start penalties added to processing times in a single facility scheduling problem, European Journal of Operational Research, 47, 56-64 (1990) · <a href="/0717.90034" class="nowrap">Zbl 0717.90034</a></td> </tr><tr> <td>[25]</td> <td class="space">Liu, P.; Yi, N.; Zhou, X., Two-agent single-machine scheduling problems under increasing linear deterioration, Applied Mathematical Modelling, 35, 2290-2296 (2011) · <a href="/1217.90108" class="nowrap">Zbl 1217.90108</a></td> </tr><tr> <td>[26]</td> <td class="space">Mazdeh, M. M.; Zaerpour, F.; Zareei, A.; Hajinezhad, A., Parallel machines scheduling to minimize job tardiness and machine deteriorating cost with deteriorating jobs, Applied Mathematical Modelling, 34, 1498-1510 (2010) · <a href="/1193.90105" class="nowrap">Zbl 1193.90105</a></td> </tr><tr> <td>[27]</td> <td class="space">Mosheiov, G., Scheduling jobs with step-deterioration: minimizing makespan on a single and multi-machine, Computers & Industrial Engineering, 28, 869-879 (1995)</td> </tr><tr> <td>[28]</td> <td class="space">Mosheiov, G., V-shaped policies for scheduling deteriorating jobs, Operations Research, 39, 979-991 (1991) · <a href="/0748.90033" class="nowrap">Zbl 0748.90033</a></td> </tr><tr> <td>[29]</td> <td class="space">Mosheiov, G., Scheduling jobs under simple linear deterioration, Computers & Operations Research, 21, 653-659 (1994) · <a href="/0810.90074" class="nowrap">Zbl 0810.90074</a></td> </tr><tr> <td>[31]</td> <td class="space">Pinedo, M., Scheduling Theory, Algorithms, and Systems (2008), Prentice Hall: Prentice Hall New Jersey</td> </tr><tr> <td>[32]</td> <td class="space">Reeves, C., Heuristics for scheduling a single machine subject to unequal job release times, European Journal of Operational Research, 80, 397-403 (1995)</td> </tr><tr> <td>[33]</td> <td class="space">Rinnooy Kan, A. H.G., Machine Scheduling Problems: Classification, Complexity and Computation (1976), Nijhoff: Nijhoff The Hague, The Netherlands</td> </tr><tr> <td>[34]</td> <td class="space">Sen, T.; Gupta, S. K., A state-of-art survey of static scheduling research involving due dates, Omega, 12, 63-76 (1984)</td> </tr><tr> <td>[35]</td> <td class="space">Sun, L.-H.; Sun, L.-Y.; Wang, M.-Z.; Wang, J.-B., Flow shop makespan minimization scheduling with deteriorating jobs under dominating machines, International Journal of Production Economics, 138, 195-200 (2012)</td> </tr><tr> <td>[36]</td> <td class="space">Wang, J. B., A note on scheduling problems with learning effects and deteriorating jobs, International Journal of Systems Science, 37, 827-832 (2006) · <a href="/1126.90347" class="nowrap">Zbl 1126.90347</a></td> </tr><tr> <td>[37]</td> <td class="space">Wang, J. B.; Cheng, T. C.E., Scheduling problems with the effects of deterioration and learning, Asia Pacific Journal of Operational Research, 24, 245-261 (2007) · <a href="/1121.90066" class="nowrap">Zbl 1121.90066</a></td> </tr><tr> <td>[38]</td> <td class="space">Wang, J. B.; Huang, X.; Wang, X. Y.; Yin, N.; Wang, L. Y., Learning effect and deteriorating jobs in the single machine scheduling problems, Applied Mathematical Modelling, 33, 3848-3853 (2009) · <a href="/1205.90137" class="nowrap">Zbl 1205.90137</a></td> </tr><tr> <td>[39]</td> <td class="space">Wang, J. B.; Jiang, Y.; Wang, G., Single-machine scheduling with past-sequence-dependent setup times and effects of deterioration and learning, The International Journal of Advanced Manufacturing Technology, 41, 1221-1226 (2009)</td> </tr><tr> <td>[40]</td> <td class="space">Wang, L. Y.; Wang, J. B.; Gao, W. J.; Huang, X.; Feng, E. M., Two single-machine scheduling problems with the effects of deterioration and learning, The International Journal of Advanced Manufacturing Technology, 46, 715-720 (2010)</td> </tr><tr> <td>[41]</td> <td class="space">Wang, J.-B.; Wang, C., Single-machine due-window assignment problem with learning effect and deteriorating jobs, Applied Mathematical Modelling, 35, 4017-4022 (2011) · <a href="/1221.90050" class="nowrap">Zbl 1221.90050</a></td> </tr><tr> <td>[42]</td> <td class="space">Wang, X.-R.; Wang, J.-J., Single-machine scheduling with convex resource dependent processing times and deteriorating jobs, Applied Mathematical Modelling, 37, 2388-2393 (2013) · <a href="/1349.90416" class="nowrap">Zbl 1349.90416</a></td> </tr><tr> <td>[43]</td> <td class="space">Wang, J.-B.; Wang, M.-Z., Minimizing makespan in three-machine flow shops with deteriorating jobs, Computers & Operations Research, 40, 547-557 (2013) · <a href="/1349.90410" class="nowrap">Zbl 1349.90410</a></td> </tr><tr> <td>[44]</td> <td class="space">Wang, J.-B.; Wang, J.-J.; Ji, P., Scheduling jobs with chain precedence constraints and deteriorating jobs, Journal of the Operational Research Society, 62, 1765-1770 (2011)</td> </tr><tr> <td>[45]</td> <td class="space">Wang, J.-B.; Ji, P.; Cheng, T. C.E.; Wang, D., Minimizing makespan in a two-machine flow shop with effects of deterioration and learning, Optimization Letters, 6, 1393-1409 (2012) · <a href="/1259.90039" class="nowrap">Zbl 1259.90039</a></td> </tr><tr> <td>[46]</td> <td class="space">Wang, J.-B.; Huang, X.; Wu, Y.-B.; Ji, P., Group scheduling with independent setup times, ready times, and deteriorating job processing times, International Journal of Advanced Manufacturing Technology, 60, 643-649 (2012)</td> </tr><tr> <td>[47]</td> <td class="space">Wang, J.-B.; Wang, M.-Z.; Ji, P., Single machine total completion time minimization scheduling with a time-dependent learning effect and deteriorating jobs, International Journal of Systems Science, 43, 861-868 (2012) · <a href="/1305.90206" class="nowrap">Zbl 1305.90206</a></td> </tr><tr> <td>[48]</td> <td class="space">Wu, C.-C.; Hsu, P.-H.; Lai, K., Simulated-annealing heuristics for the single-machine scheduling problem with learning and unequal job release times, Journal of Manufacturing Systems, 30, 1, 54-62 (2011)</td> </tr><tr> <td>[49]</td> <td class="space">Yang, S.-J.; Yang, D.-L., Minimizing the makespan on single-machine scheduling with aging effect and variable maintenance activities, Omega, 38, 528-533 (2010)</td> </tr><tr> <td>[50]</td> <td class="space">Yin, Y.; Xu, D., Some single-machine scheduling problems with general effects of learning and deterioration, Computers and Mathematics with Applications, 61, 100-108 (2011) · <a href="/1207.90060" class="nowrap">Zbl 1207.90060</a></td> </tr><tr> <td>[51]</td> <td class="space">Zhao, C.; Tang, H., Rescheduling problems with deteriorating jobs under disruptions, Applied Mathematical Modelling, 34, 238-243 (2010) · <a href="/1185.90114" class="nowrap">Zbl 1185.90114</a></td> </tr><tr> <td>[52]</td> <td class="space">Zhao, C.; Tang, H., A note to due-window assignment and single machine scheduling with deteriorating jobs and a rate-modifying activity, Computers and Operations Research, 39, 6, 1300-1303 (2012) · <a href="/1251.90209" class="nowrap">Zbl 1251.90209</a></td> </tr><tr> <td>[53]</td> <td class="space">Zhao, C. L.; Zhang, Q. L.; Tang, H. Y., Scheduling problems under linear deterioration, Acta Automatica Sinica, 29, 531-535 (2003) · <a href="/1498.90109" class="nowrap">Zbl 1498.90109</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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